You have found the following ages (in years) of all 6 sloths at your local zoo: $ 14,\enspace 21,\enspace 11,\enspace 10,\enspace 2,\enspace 18$ What is the average age of the sloths at your zoo? What is the variance? You may round your answers to the nearest tenth.
Because we have data for all 6 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $6$ ages and divide by $6$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\mu} = \dfrac{14 + 21 + 11 + 10 + 2 + 18}{{6}} = {12.7\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $14$ years $1.3$ years $1.69$ years $^2$ $21$ years $8.3$ years $68.89$ years $^2$ $11$ years $-1.7$ years $2.89$ years $^2$ $10$ years $-2.7$ years $7.29$ years $^2$ $2$ years $-10.7$ years $114.49$ years $^2$ $18$ years $5.3$ years $28.09$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{1.69} + {68.89} + {2.89} + {7.29} + {114.49} + {28.09}} {{6}} $ $ {\sigma^2} = \dfrac{{223.34}}{{6}} = {37.22\text{ years}^2} $ The average sloth at the zoo is 12.7 years old. The population variance is 37.22 years $^2$.